Travis told CNBC that delivery constituted "the strategic plank to many things I'm interested in" such as curbside service, he added.

The British- born CEO of Dunkin' Brands has run the company, which includes Baskin-Robbins in addition to the Dunkin' Donuts brand, since January 2009. He sees delivery as the next frontier in serving consumers that are more technology savvy than ever, and expect service providers to keep pace.

"The trend in this country is convenience. So I think delivery plays very strongly," Travis said. "I think the next few years you're going to see us get more and more into delivery."

But Travis cautioned that the company, which has more than 8,000 U.S. locations, won't rush into a full launch.

"The key thing is we have to make sure we can operationalize all these procedures. That's why I've slowed up mobile ordering to make sure it's operationally simple at the store level."

Last year, Dunkin' posted disappointing sales, amid stiff competition among restaurant chains and an environment where consumers are still struggling economically.

When asked about the economy, Travis tells CNBC he thinks its "doing better than many people think. "That's based on our franchisees talking about finding labor. It's getting more and more difficult."

He adds, "I actually feel pretty good about the US economy right now. And probably I feel better than I did nine months or a year ago."

Getty Images

People wait in line in the morning at a Dunkin' Donuts chain restaurant in New York City.

In the coffee and bakery segment, industry leader Starbucks commands a 35 percent market share. However, Dunkin' Donuts is second with 19.8 percent share, according to marketing research from IBISWorld. Other players include Tim Hortons and Krispy Kreme with a 1.9 percent and 1.6 pecent market share, respectively.

"We always talk about Starbucks and McDonalds, but there [are] a lot of independents out there."

According to IBISWorld, more than 40 percent of the players in the category are small competitors.

"There's a lot of competition," the Dunkin' Donuts CEO said, "one of the things I like about our business is you wake up every day and you're competing."

On the Money airs on CNBC Sundays at 7:30 pm, or check listings for airtimes in local markets.